Manuscript X-C-ST@1 · X-C-ST.tex · section · b679954e

Formal Green operators

Yang–Feldman characterization and quantum order › Formal Green operators · lines 936–965

X-C-ST@1#eq:perturbed-green

Define, coefficientwise through $κ^{2}$ ,

E_{q}^{R} E_{q}^{A} = (1 + E_{R} L [q])^{- 1} E_{R} = E_{R} (1 + L [q] E_{R})^{- 1}, = (1 + E_{A} L [q])^{- 1} E_{A} = E_{A} (1 + L [q] E_{A})^{- 1} = E_{A} R_{q} .

For instance,

E_{q}^{A} = E_{A} - κ E_{A} L_{1, g} E_{A} + κ^{2} (E_{A} L_{1, g} E_{A} L_{1, g} E_{A} - E_{A} L_{2, g} E_{A}) .

These are finite formal compositions, not numerical inverses of a background-dependent operator at finite coupling.

Let

ϕ_{q} (c) = ϕ (R_{q} c) .

This is a linear formal field in the inherited free algebra. No state is part of this definition.

Displayed formulas in this passage

X-C-ST@1#eq:perturbed-green · line 939
$E_{q}^{R} E_{q}^{A} = (1 + E_{R} L [q])^{- 1} E_{R} = E_{R} (1 + L [q] E_{R})^{- 1}, = (1 + E_{A} L [q])^{- 1} E_{A} = E_{A} (1 + L [q] E_{A})^{- 1} = E_{A} R_{q} .$
unlabeled · line 950
$E_{q}^{A} = E_{A} - κ E_{A} L_{1, g} E_{A} + κ^{2} (E_{A} L_{1, g} E_{A} L_{1, g} E_{A} - E_{A} L_{2, g} E_{A}) .$
unlabeled · line 960
$ϕ_{q} (c) = ϕ (R_{q} c) .$